Sample moments

Summary

Definitions

  • A population is a set of similar items such as all individuals in Sweden or all firms in Europe.
  • A sample is set of data selected from a population by some defined procedure.
  • A sample is a random sample if each item in the population has the same probability of be selected into the sample.
  • Each element of the sample is called a sample point . The symbol \(n\) is the common symbol for the size of the sample, that is, the number of sample points.
  • \(x_i\) will denote a particular measurement of sample point \(i\) , for example the income of individual \(i\) , where \(i\) is any number from 1 to \(n\) .
  • We often refer to the \(n\) numbers \(x_1,x_2,…,x_n\) as the sample.

Sample mean, sample variance and sample standard deviation

\[\bar{x}= \frac{1}{n}\sum_{i=1}^{n}{ x_i }\]

  • The sample variance

\[s_x^2= \frac{1}{n-1}\sum_{i=1}^{n}{ {\left( x_i-\bar{x} \right)}^2 }\]

  • The sample standard deviation

\[s_x=\sqrt{s_x^2}\]

Sample covariance and sample correlation

  • Given a population, we can make two distinct measurements from sample point \(i\) and denote them by \(x_i\) and \(y_i\) . For example, \(x_i\) could be the total revenue and \(y_i\) the total cost for firm \(i\) .
  • Given data \(x_1,x_2,…,x_n\) and \(y_1,y_2,…,y_n\) we define
  • The sample covariance

\[s_{x,y}^2= \frac{1}{n-1}\sum_{i=1}^{n}{ \left( x_i-\bar{x} \right)\left( y_i-\bar{y} \right) }\]

  • The sample correlation

\[r_{x,y}^2= \frac{s_{x,y}^2}{s_xs_y}\]